Analysis of the Gas Diffusion Layer in a PEM Fuel Cell Teng Zhang, E.Birgersson

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1 Analysis of the Gas Diffusion Layer in a PEM Fuel Cell Teng Zhang, E.Birgersson Engineering Science Programme, Faculty of Engineering, National University of Singapore Kent Ridge Road, Singapore 7546 ABSTRACT In this study, a pore network model of the gas diffusion layer (GDL) is constructed by idealizing the GDL as a 2D lattice comprised of pore bodies and pore throats. Geometric parameters and transport properties of the model are compared with experimental values. By solving incompressible Navier-Stokes equations and Fick s law of diffusion on various models, absolute permeability and effective gas diffusivity of GDL with different porosities and structures are obtained and compared. Finally, possible methods of studying two phase flow problems with pore network model are discussed. INTRODUCTION In view of the increasing level of environmental pollution and global energy consumption, fuel cells and have emerged as a promising alternative power plant for automotive, stationary and portable applications. Due to high efficiency, low emission, low noise and a low operating temperature, development of proton exchange membrane fuel cells (PEMFC) has been accelerating since the late 99s []. The performance of a PEMFC is largely affected by the multiphase transport phenomena in the gas diffusion layer (GDL). GDL typically has a fibrous structure, in the form of carbon paper or carbon cloth. It plays multiple roles in the operation of PEMFC: its high porosity and hydrophobicity allows diffusion of the reactant gas and discharge of the produced water through the pores; its low electronic resistively also allows collection of current through the fibres. A micro-porous layer consists of carbon powder and hydrophobic agent is often inserted between catalyst layer and GDL to improve water management and gas diffusion capacity. A schematic of cathode GDL in between flow field and PEM is shown in Fig.. Conventional method of modelling multiphase flow in GDL has been based on continuum description of the flow and transport. However, due to the lack of experimental data, there are a number of problems with the constitutive relations used in such models. The constitutive relation for capillary pressure as a function of saturation has been so far relied on Leverett J function, which is approximated based upon the experimental data obtained with soil samples [2]. To get more reliable constitutive relations, a detailed study on the morphology of GDL is required. A number or works have been done in this field [3-5], including the reconstruction of voxel image of GDL form micro focal X-ray slice images. Numerical works with detailed GDL structure implemented would be more reliable. However, solving continuity and conservation equations on such complex geometries is computationally intensive.

2 Fig. Schematic of PEMFC cathode Fig 2. SEM surface image of Toray 9 An alternative way of studying multiphase flow in GDL is pore network modelling, which is originated from geological science [6]. Earlier work which applies pore network modelling in studying GDL is done by Thompson [7], and a number studies with different emphasis followed [8-9]. The idea of pore network modelling is to represent the complex geometry of GDL by a collection of wide pore bodies communicating through narrower regions called throats. The relatively simple geometry of pore network model makes it easy to study and calibrate. Pore network modelling is especially simple and effective in studying two phase flow process governed by pore-level physics. In this paper, a 2D pore network model is constructed. It is then utilized in single phase flow simulations to investigate the influence of porosity and pore structure on GDL s absolute permeability and effective gas diffusivity. The work shows that pore network model is a relatively simple and effective method in studying properties of GDL. Nomenclature b min minimum pore radius (m) P pressure (Pa) b p, i radius of the ith pore (m) q Darcy flux (m/s) c concentration (mol/m 3 ) u velocity (m/s) D plain diffusivity of air (m 2 /s) α e Effective diffusivity of PEM D eff effective diffusivity (m 2 /s) χ, χ max random number in (,) F Faraday constant (96487C/mol) ε porosity J current density (A/m 2 ) γ surface tension (N/m) K absolute permeability (m 2 ) κ parameter in Weibull distribution L c lattice constant (m) λ parameter in Weibull distribution L t throat length (m) ρ w density of water (997kg/m 3 ) M air molar mass of air (29g/mol) θ contact angle M water molar mass of water (8g/mol) µ dynamic viscosity of water N flux (mol/m 2 s) (8.9-4 Pa.s)

3 MODELLING PROCESS Model Construction Construction of the first pore network model is based on the physical properties of carbon paper Toray 9, which are listed in Table. Toray 9 consists of nonwoven linear fibres mostly arranged in layers in the plane of paper. A SEM surface image of Toray 9 is shown in Fig.2. Table. physical properties of Toray 9 Property Thickness Porosity In plane permeability Through plane Permeability Value 29 µm m m 2 The model developed here simplifies the one described by Fowler [9] into a 2D lattice. The construction of the model begins with assigning each site of the lattice a rectangular square representing the pore body. These pores bodies are then connected by ducts of square section called throats. The arrangement is shown in Fig.3. Rectangular shape is used because it is easy Fig.3 Schematic of pore-throat construction Fig. 4 Voxel images and pore network models to construct, and it qualitatively describes the presence of corners and crevices in the pore space [9]. The distance between adjacent lattice sites is called lattice constant (L c ) and it is carefully adjusted according to total porosity of the model and pore and throat size distributions. In the present model, L c is taken to be 25 µm. Pore size distribution of the model is assumed to follow a truncated Weibull cumulative distribution [9] give by: b p, i = λ[ ln ( χχ max )] /κ + b min () Advantage of this distribution is that it allows easy modification and calibration of the model. Once pore sizes are assigned, throat width is taken as the diameter of the smaller one of the two adjacent pores. Throat length (L t ) is then obtained by subtracting radiuses of the two adjacent pore bodies from L c. The result of such arrangement is pores and throats having similar size,

4 which allows minimum constriction between the pores. This property of the model maps the highly open structure characteristic of GDL. Topological study of Toray 9 shows that fibres are aligned in in plane direction other than through plane direction, resulting in different permeability in these two directions (see Table ). To account for this fact, pore distributions are roughly correlated in the model. The correlation makes pores of similar size more likely to be aligned in in plane direction other than in through plane direction. Model Variation Six different models are built based on the method described in the last section. The first four models have different pore body size distributions corresponding to GDL with different porosities. The details of the first four models are listed in Table 2. Note that the third model is the one representing Toray 9. Fig.4 shows voxel images of GDLs of different porosities and their corresponding pore network models. The fifth model assigns a linearly increasing porosity in through plane direction and it has an overall porosity of.8. It is shown in Fig.5. In the last model, a MPL with.52 porosity and 5µm thickness is attached at one end of the third model. Like GDL, the MPL is represented by a pore network model. Table 2. Parameters used for the first four models Fig.5. Model with linearly varying porosity Model number Porosity Parameter λ Parameter κ Numerrical Simulations Absolute permeability. The domain of interest is a pore network model L =29µm long in z direction and L =32µm long in y direction. The process to be simulated is discharge of water through the network model. Water is assumed to be injected at the left boundary at z=. The flow of water inside the model is governed by the incompressible Navier-Stokes equations: u ρ µ t u= u+ 2 w ρ w ( u ) u+ p= (2) Gravity force has been neglected. At z=, inward velocity boundary condition is used: u =u (3)

5 A current density of A/m and an effective diffusion coefficient of.5 of the PEM are assumed during operation of the cell. So Darcy flux q is given by: JM water q = (+ 2α e ) = u 2 Fρ w The right boundary at z = L is set as a pressure boundary: (4) P L = (5) The average pressure at the left boundary can be evaluated by integrating p over the boundary and then divide by the length of the boundary: L P avg = pdy (6) L Total Dacry flux at the right boundary z=l can be found by integrating Darcy flux over the boundary: Darcy flux at this boundary is related with velocity by: L Q= q dy (7) q= εu (8) Whereε is found by integrate over the whole domain. Once Q and P avg is known, absolute permeability can be found from Darcy s law: KL Q= ( P avg P L ) (9) µ L Effective gas permeability. The domain of interest is the same as in the last section. Gas diffuses from the right boundary to the left boundary. The diffusion of gas inside the model is governed by Fick s law of diffusion: c + ( D c) = t () The left boundary is set as a flux boundary, given by: J ( D c) n = N = L M air 2F ()

6 Average flux at this boundary is then: Similarly, average concentration is given by: L L N avg = N dy (2) L L c avg = cdy (3) L The left boundary is taken as a concentration boundary: c = c = -5 mol/m 3 (4) The effective diffusivity of the model can be calculated from: c eff avg D = L c N avg (5) RESULTS AND DISCUSSIONS Pore size distributions Pore size distributions of the first four models with parameters listed in Table 2 are shown in Fig. 6. Average pore diameters are listed in Table 3. Experimental results from Schulz et al. [5] have shown the average pore diameter of Toray 9 to be within 9~2µm, which confirms with the third model having an average pore diameter of 9.43µm. However, it can be seen from the figure that the spread of the distribution curves increases while the peak value of the curves decreases with porosity. This suggests that either parameter λ is too large or parameter κ is too small. A more elaborate way of constructing these models is adjusting parameters λ, κ and b min as well as the value of L c, which is fixed in present models. Table 3. Average pore radius Model porosity Average pore radius (µm) Fig.6. Pore size distributions

7 Absolute permeability Absolute permeability values of all six models are listed in Table 4. A plot of permeability values of the first four models against porosity is shown in Fig. 6. The experimental permeability value of Toray 9 is 9. in through plane direction and 5. in in plane direction [9]. This is quite close to the numerical value obtained with the third model. Results for the model having a linearly decreasing porosity are obtained by running the simulation with the fifth model but having water injected at z = L instead of z =. It can be seen that models with linearly varying porosity have smaller absolute permeability values than the normal one with the same overall porosity. The model with the effect of MPL considered also has smaller absolute permeability compared with the normal one. This is predictable as the MPL drugs down the overall porosity. However, the value of absolute permeability does not reflect material properties of the MPL which can affect contact angle of the liquid. Contact angle is an important parameter that affects the effective permeability in multiphase flow process. Thus, we do not expect water to be discharged less efficiently with MPL by judging from the decreased absolute permeability of the model. Fig.6 through plane permeability vs porosity Fig.7 effective gas diffusivity vs prorsity Table 4. permeablility of each model Model Through plane permeability(m 2 ) In plane permeability(m 2 ).62 porosity porosity porosity porosity linearly increasing porosity linearly decreasing porosity porosity with MPL Fig.6 indicates an exponential relationship between plane permeability of the model and porosity. This relationship can be very well approximated by the curve: K ε = 2 e (6)

8 Effective gas diffusivity Effective diffusivities of all models are listed in Table 5. Fig.7 plots effective diffusivities of the first 4 models against porosity. There is currently no experimental data available for effective diffusivity of Toray 9, so these values cannot be validated. The first four models give very close results, implying porosity does not affect the effective gas diffusivity significantly. It is noticeable that the model with linearly increasing porosity has much a larger effective diffusivity, while the one with linearly decreasing porosity has a smaller effective diffusivity, compared with the normal one. The model with MPL added has a slightly higher effective gas diffusivity. It can be seen that structure of the GDL affects its effective gas diffusivity in a quite complex way. Table 5. effective diffusivity of each model Model Effective gas diffusivity (m 2 /s).62 porosity porosity porosity porosity linearly increasing porosity linearly decreasing porosity porosity with MPL.4-5 Fig.7 shows a rough logarithmic relationship between effective gas diffusivity and porosity. The approximated relationship is given by: = ε + eff 6 6 D 3 ln( ) (7) The approximation is not very good, since the interception indicates non-zero diffusivity at zero porosity, which clearly doesn t make sense. CONCLUDING PROSPECTIVE The study of absolute permeability and effective gas diffusivity of GDL using pore network models has shown some interesting results. With exceedingly simple geometry and minimal amount of computation, pore network modelling predicts an exponential dependence of absolute permeability of the GDL on its porosity, and points out the complex nature of the relationship between structure of the GDL and its effective gas diffusivity. For further work, models with a variety of different structures could be built to explore and investigate the optimal structure for effective gas diffusivity. With single phase flow problems studied by pore network modelling, the next logical step is to apply pore network models to study multiphase transport problems. A direct approach, which can be regarded as an extension of the methods studied in this paper, would be applying level set method described in [] to pore network models. Level set method solves incompressible Navier-Stokes equations and diffusion equations for two species with a proper level set equation

9 incorporated. By applying such method, flow patterns within the model and the behaviour of gas-water interfaces can be visualized and studied. However, since information such as material properties and small-scale geometric characterizations of GDL are not reflected in pore network models, solve continuum equations on the model will not give accurate results. A more interesting application of pore network models on two phase flow studies is simulation of drainage and imbibition processes in GDL. The simulation is based on an invasion percolation algorithm [9, ]. For such simulations, it is assumed that the flow and transport behaviours in GDL is governed by capillary pressure, which is defined as the pressure difference between the non-wetting fluid (water) and the wetting fluid (gas). The value of capillary pressure is quantitively described by the Young-Laplace equation: 2γ cosθ Pc= (8) b t where γ is the surface tension, θ is the contact angle and b t is the radius of the pore or throat. Consider, for example, a drainage process. The algorithm first assigns a capillary pressure at every throat according to (8). Then, a low pressure is given at the injection boundary and all throats that can be penetrated at the given pressure (that is, P c <P) are marked as open. Next, all clusters formed by open throat and the pore bodies connected to them are identified. Finally, the identified clusters that are connected to the injection boundary are invaded. The process goes on with P c increased by a small amount on each step. For incompressible fluid, if the invading fluid surrounds a blob of the displaced fluid, the blob is kept intact for the rest of the simulation. The process stops when all the pores and throats are filled with invading fluid. Fig.7 A pattern of invasion percolation This process is conceptually clear and easy to simulate. A visualization of invasion percolation process is shown in Fig. 7. The simulation can be used to study water distributions during a drainage process inside the GDL and a capillary pressure against saturation curve can be obtained after the simulation. By incorporate conservation equations and other constitutive relations into the simulation, the relationship between relative permeability and saturation can also be studied. It is seen that pore network modelling has a lot of potential in studying two phase flow problems. Its simplicity and effectiveness make it a valuable tool in investigating GDL performances. REFERENCES [] G. Hoogers, Fuel Cell Technology Handbook, CRC Press LLC, 22. [2] N. Djilali, Energy 32 (27)

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